Delft University of Technology
A Quasi-Analytical Tool for the Characterization of Transmission Lines at High
Frequencies
Berkel, S. van; Garufo, A.; Llombart Juan, N.; Neto, Andrea
DOI
10.1109/MAP.2016.2541617
Publication date
2016
Document Version
Accepted author manuscript
Published in
IEEE Antennas and Propagation Magazine
Citation (APA)
Berkel, S. V., Garufo, A., Llombart Juan, N., & Neto, A. (2016). A Quasi-Analytical Tool for the
Characterization of Transmission Lines at High Frequencies. IEEE Antennas and Propagation Magazine,
58(3), 82-90. https://doi.org/10.1109/MAP.2016.2541617
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A Quasi-Analytical Tool for the Characterization
of Transmission Lines at High Frequencies
Sven L. van Berkel, Alessandro Garufo, Nuria Llombart, and Andrea Neto
I
n this article, we present a freely accessible software tool that allows for fast characterization of dynamic phenomena in a wide variety of trans-mission lines that include character-istic impedance, effective dielectric constant, and losses, such as radia-tion into space and surface waves. For printed transmission lines, the radia-tion effects are of particular impor-tance when the transverse dimensions of the transmission lines become sig-nificant in terms of wavelength. Gen-erally, dispersion and losses of the line due to these dynamic phenomena are predicted by full-wave simulations as quasi-static formula do not suffice. The presented software tool, freely downloaded from http://terahertz. tudelft.nl, is capable of accurately ana-lyzing the most widely used transmis-sion lines at high frequencies.At low frequencies, for example, in the case of printed transmission lines below ~50 GHz, the main parameters of a transmission line, such as the propaga-tion constant, characteristic impedance, and attenuation constant, can generally be approximated using quasi-static formulations [1]. However, when the transverse dimensions of the transmis-sion lines become significant in terms of the wavelength (+m/ ),20 dynamic
phenomena in the line become nonnegli-gible and can have a significant influence on these parameters. The excitation of leaky higher-order modes causes radia-tion into space and surface waves, which launches power within the stratification [2], [3]. In fact, multiple modes can prop-agate simultaneously in the transmis-sion line [4]. These phenomena could be avoided with micrometric integrated technology. However, one may wish to still use low-cost printed circuit board
technology while minimizing the effects of these higher-order modes. For exam-ple, a coplanar waveguide (CPW), with a 100- mn minimum feasible dimension in width and spacing (see Figure 1), will radiate while excited in its differential propagation mode at higher frequencies (f$50 GHz). Moreover, these effects will also be present in integrated tech-nology circuits operating in the submil-limeter-wavelength range. Therefore, it is of importance to characterize the
editor’s Note
High-frequency transmission lines bring a variety of challenges, both practical and theoretical. Regarding the latter, quasi-static approaches are generally of limited use because one has to deal with a variety of full-wave effects. This issue’s “EM Program-mer’s Notebook” column provides a review of the issues that must be addressed and presents a publicly available software tool.
d H ws y z x εr
FIGURE 1.
An example of a transmission line (CPW) with its reference axis. ws isthe width of the slot (or strip for strip-type structures), d is the spacing between multiple lines, and H and er are the height and the relative permittivity of the
impact of these dynamic effects. To date, there are no (quasi)analytical tools for estimating radiation losses. In addi-tion, equivalent formulas for coplanar transmission line surface-wave losses [2] are not applicable for more complex structures dealing with an arbitrary stratification or superconductivity. A designer willing to perform a detailed analysis will be obliged to resort to the use of full-wave simulations that are very time consuming and require expensive licenses.
For this purpose, a freely available software tool (Figure 2) has been devel-oped that can characterize transmission lines at high frequencies. This tool is based on a fast quasi-analytical model that is more extensively described in [5]–[8].
QUASI-ANALYTICAL MODEL
The tool follows a quasi-analytical approach that makes use of transmis-sion line formalism [6]–[8]. This for-malism derives the Green’s functions (GFs) of the transmission line by solv-ing the pertinent integral equation of infinite line currents radiating in the presence of stratified media, excited by a 3 gap (Fig ure 3). In the software tool,
four possible layers of stratification are considered:
1) an infinite top medium 2) a finite upper slab 3) a finite lower slab
4) an infinite bottom medium. With this choice of stratification, most commonly used printed transmission lines can be modeled having a different number of conductors (e.g., see Figure 1).
TRANSMISSION LINE FORMALISM
In the setup of the problem, the trans-mission line is composed of n guid-ing structures (strips/slots), e.g., n = 1 for a microstrip and n = 2 for a CPW. The transmission line is assumed to be oriented along ,xt the conductors are
infinitesimal in thickness, and the strati-fication is homogeneous in the ( , )x yt t
plane. The transmission line formalism is constructed by first defining two types of integral equations: the elec-tric field integral equation (EFIE) for strips in [6] and [7] or the continuity of mag-netic field integral equation (CMFIE) for slots in [8]. In this formulation, the elec-tric (magnetic) fields are averaged over the width of the strip (slot). The unknown equivalent elec-tric ( magnetic) current distributions along the lines c x yeq( , ), are assumed to be separable in space dependency [5]:
( , ) ( ) · ( )
c x yeqi =c x c yi t -idy for i = 0:
n-1, where dy=d w+ s is the
spac-ing between the centers of the copla-nar lines (Figure 1). The width of the line is assumed to be much smaller than the wavelength ws%m, which allows for characterizing the transverse dependence ct(y) by the quasi-static
edge singularities: . w 1 ( ) y c y w w y 2 1 2 1 for s t s s 2 r = -c m
In this way, ( )c x represents the
longi-tudinal electric currents (voltage drop) along the strips (slots).
After representing the pertinent inte-gral equations in the spectral domain and equating the integrands, the lon-gitudinal electric or magnetic currents
( )
c x along the transmission line can be
expressed as an inverse Fourier trans-form of the current spectrum in the lon-gitudinal domain of the lines (1). This formulation is known as transmission
line formalism: ( ) ( ) ( ( )) ( ) , sin det sin c c c x D k N k e dk D k A k N k e dk 21 2 21 2 x x jk x x x x x jk x x 1 0 0 x x # # r r D D = = 3 3 3 3 -c c m m
#
#
(1) where ( )A kx is the adjugate of ( ).D kxFor the EFIE (CMFIE), the denomi-nator D k( )x represents the average
transverse electric (magnetic) field radi-ated on the strip (slot) by the equivalent
FIGURE 2.
The GUI of the software tool using the presented quasi-analytical model. The most commonly used printed transmission lines, with user-defined stratification, can be simulated.We present a freely
accessible software
tool that allows for fast
characterization of dynamic
phenomena in a wide variety
of transmission lines.
currents. The excitation law N0 of the magnetic (electric) dipole sources in the 3 gap (Figure 3) must ensure that the transmission line is fed by 1 V (1 A) in the case of strips (slots). This implies that the amplitude of the excitations |N0,i+1|=1/n (V or A) for i=0:n-1.
Moreover, N0 has to select any propa-gation modes (e.g., common or differ-ential excitation) for transmission lines composed of n2 guiding structures, 1 e.g., for a CPW excited in its differential propagating mode N0= [0.5 – 0.5] A. The denominator ( )D kx can, in turn,
be expressed as a transverse spectral integration (2): ( ) ( , ) ( ) D k G k k C k e dk 21 , , ( ) n i x xxF C x y t y jk n i d y 1 y y # r = 3 3 + --
-#
(2) for i=0:n-1. In (2), G , ( , )k k xx F C x y isthe spectral GF of the correspond-ing stratification in the absence of the guiding structure: GE Jxx,( , )k kx y for
strip-type and G , ( , )k k
xx H M
x y for
slot-type transmission lines. The GF of this stratification can be modeled as planar stratified media [9]. The assumed edge singular distribution ct(y) has an
ana-lytic Fourier transform .
J k w
2
y s
0c m
The averaging of the fields over the width of the strips (slots) leads to a sinc-function multiplication of the transverse current distribution:
( )
C kt y =J0ck wy2 sm sinc ck wy2 sm.
The exponential term in (2) accounts for the coupling between multiple lines.
INTEGRATION PATH
The result of the integration in kx from
–∞ to +∞ in the inverse Fourier trans-form (1) depends mostly on the polar singularities in D-1( ).kx The procedure
of finding these polar singularities is based on solving det D k( ( ))x =0, which defines the dispersion equation. These
singularities are associated with the propagating modes along the transmis-sion line. Three characteristic propagat-ing modes are of interest:
1) bounded modes: nonradiative modes such as the main propagating mode in a microstrip
2) surface-wave modes: propagating modes exciting a surface wave intrin-sic to the stratification
3) leaky-wave modes: propagating modes radiating in a dense infinite medium, as in the case of circuits placed at the bottom of a dielectric lens.
To evaluate the singularities of ( ),
D kx the integration in the transverse
spectral domain (2) must be performed numerically. However, the choice of the transverse integration path in ky
is nontrivial because it depends on the location of the propagating mode in the
longitudinal kx domain. When an
infi-nite stratification is present, branch cuts will appear in the spectrum, leading to different Riemann sheets. Changing the integration path into the different Rie-mann sheets allows for finding not only bounded modes but also leaky modes. The integration path of the bounded- and surface-wave mode for a microstrip has been extensively studied by Mesa [6], [7], and a leaky-wave mode of a slot-line has been studied by Neto [8]. In choosing the integration path, care has to be taken in guaranteeing that the solu-tions of the dispersion equation are actu-ally physicactu-ally valid propagating modes. For example, when a surface-wave pole is present and excited, this pole should be enclosed by the transverse integra-tion path. A surface-wave pole is excited when bmode1bSFW [2], where bmode
and bSFW are the phase constants of
the main propagating mode and the surface-wave pole, respectively. In addi-tion, when the main propagating mode is in fact a leaky-wave mode, the trans-verse integration path in (2) should cross
branch cuts to integrate over the bottom Riemann sheet, where the leaky-wave pole can be found.
Using the software tool, the height of the stratification is limited such that only one mode will propagate in the transmission line. The tool will automati-cally select the appropriate integration path associated with this specific type of propagating mode.
TRANSMISSION LINE CHARACTERISTICS
Using the quasi-analytical model described in the previous section, numerous interesting characteristics of transmission lines can be analyzed. The model is used to generate a software tool that is made freely accessible and capa-ble of analyzing the most widely used printed transmission lines. The graphical user interface (GUI) is shown in Fig-ure 2. The user can select the materials and loss tangents of four possible lay-ers of stratification: two finite dielectric slabs and two infinite media. Conduc-tor and dielectric losses in the structure are described by supplying a value for
FIGURE 3.
The Δ-gap excitation for (a) strip-type and (b) slot-type transmission lines, a magnetic dipole mΔ(x, y), and an electric dipole jΔ(x, y).∆ ∆ (a) (b) ws ws m∆ j∆
To date, there are no (quasi)
analytical tools for estimating
radiation losses.
a finite conductivity and loss tangent, respectively. The tool can characterize radiation losses, and it takes account of the first surface wave appearing in the stratification. The output of the tool is composed of four values, as can be seen in Figure 4:
■ the complex normalized wavenumber ■ the effective dielectric constant ■ the characteristic impedance ■ losses.
This section describes the imple-mentation and validation of these transmission line characteristics in the discussed quasi-analytical model. The transverse dimensions ws and d of the
structures to be validated are gener-ally chosen to be the minimum trans-verse dimensions typically possible for printed circuit board technology, i.e., 100- mn strip (slot) width and 127- mn dielectric slab height. A CPW will be excited with a -3gap excitation in the differential mode.
The tool solves the dispersion equa-tion by using a Taylor series expansion around an initial guess point kinit for the
propagation constant along the line. This leads to a first-order approximation of the propagation constant associated to the main mode (3):
e i i i ( ( )) ( ( )) . det det kmod kin t D kD k in t in t . - l (3)
For n coplanar lines, the dispersion equation will have n solutions, corre-sponding to the number of supported propagating modes along the line. The tool allows for selecting the propagat-ing mode of interest (e.g., differential or common modes in a CPW). The first-order Taylor expansion is repeated by the algorithm until the achievement of a con-vergence goal that can be specified by the user. The resulting normalized com-plex wavenumber k∕k0, with k=b-ja
and k0 the free-space wavenumber, gives
information regarding the dispersion of the transmission line and the losses asso-ciated with it.
The capability of characterizing this dispersion and losses is one of the sig-nificant advantages of using the pre-sented quasi-analytical model. Losses due to radiation into space and surface
waves in transmission lines with mul-tiple and arbitrary layers of stratifica-tion can be easily investigated. In the remaining part of this section, the main parameters of a printed transmission line are investigated and validated when such dynamic phenomena are present and quasi-static formula often do not suffice. In these cases, the designer wishing to estimate these phenomena is generally required to resort to the use of full-wave simulations. In the pre-sented validation, losses due to radia-tion into space and surface waves are studied, as well as ohmic losses in the conductors or stratification and the use of superconductors.
PHASE CONSTANT
The real part of the complex wavenumber b is the phase constant of the main propa-gating mode. The phase con-stant allows for calculating the effective dielectric con-stant of the transmission line
. , r 0 2 eff bb e =c m
In Figure 5, one can see the real par t of the complex normalized wavenumber ( / ),b b0 for a microstrip
with ws=100 mn , H= 127 mn , and . .
11 9
r
e = Also shown is a CPW with
,
ws= 100 mn d=100 mn , H=3, and er=11 9. . The presented results are validated with full-wave simula-tions done in CST, where the main line parameters are extracted from the field distribution along the strip and slots.
ATTENUATION CONSTANT
The imaginary part of the wavenum-ber a is the superposition of attenuation due to conductor, dielectric, and radia-tive losses. These radiation losses can
20 40 60 –1 0 1 2 3 Frequency (GHz) (a) 20 40 60 Frequency (GHz) (b) 20 40 60 Frequency (GHz) (c) 20 40 60 Frequency (GHz) (d) kmode /k0 Propagation Constant –20 0 20 40 60 Z0 (Ω ) Characteristic Impedance 10 20 30 40 50 (dB/m) Attenuation 7.4 7.5 7.6 7.7 7.8
7.9 Effective Dielectric Constant Re(k) Im(k) kTM0 Re Z0 Im Z0 εeff
FIGURE 4.
The output of the software tool: (a) complex wave number, (b) characteristic impedance of the main propagating mode, (c) losses (asuperposition of ohmic and radiative losses), and (d) effective dielectric constant.
Losses due to radiation
into space and surface
waves in transmission lines
with multiple and arbitrary
layers of stratification can
be easily investigated.
be due to excitation of an intrinsic sur-face wave or direct radiation into a dense infinite medium such as a silicon lens. The attenuation constant ,a in nepers, can be converted to a loss figure, e.g., in
/ , dBmeff where f cr, eff eff m e =
is the effective wavelength of the main propagating mode.
DIELECTRIC LOSSES
Besides conductor losses, transmission lines can suffer from dielectric losses. These losses can be defined in the tool by means of a dielectric loss tangent
( )
tan d. This loss tangent is implemented
in the GF of the stratified media by defining the dielectrics with a complex relative permittivity (4): (1 jtan( )) r 0 lossy e d e =e - . (4)
The d ielectr ic losses for a microstrip and CPW are validat-ed with ws=100 mn , d= 100 mn ,
,
H=127 mn a nd er= 11 9. . T he dielectric slabs are characterized with a loss tangent of tand=0 005. . The results are shown in Figure 6 and are validated with CST.
CONDUCTOR LOSSES
Using the software tool, conductors are modeled to be infinitesimal. However,
a change in surface impedance due to the finiteness of the conductor is imple-mented using the surface-impedance given in (5) [10]: , Z e R 1 s RR RF DC RF = - - (5) where RDC= v1t and ( ) RRF= 1+j rnv0f
with thickness t, conductivity ,v and free-space permeability .n0 Ohmic
loss-es in ground planloss-es for both slot- and strip-type transmission lines can be eas-ily implemented in the GF of the strati-fied media using a load that is equal to the high-frequency surface impedance
Zgnd = RRF. The conductor losses for the
main conductor of a strip-type transmis-sion line are accounted for by means of a surface impedance boundary condition; imposing the EFIE on a lossy conduc-tor gives rise to a nonzero tangential total electric field contribution. The total electric field etot can be related to the
strip’s surface impedance and current along the line: etot=Zstrip( )y c x yeq( , ). It makes sense to define Zstrip(y) as a
function of the surface impedance Zs.
Proceeding as described in [10] and [11], accounting for the ohmic losses in the metal leads to a new denominator for the strip (6): ( ) ( ) . D k D k w2 Z I x x s s lossy = + r (6)
In (6), I is the identity matrix. The factor /w2 sr accounts for the
trans-verse edge singular current distribu-tion [11]. Because the conductors are modeled to be infinitesimal, the electric current in the EFIE flowing on the top of the conductor is considered to be equal to the current flowing on the bottom of the conductor. However, for a microstrip on top of a dense dielectric slab, the current will mainly flow on the bottom of the conductor. This will have its influence on the effective surface impedance of the conductor. This cur-rent ratio can be approximated by mak-ing an assumption of the change in field
FIGURE 6.
Dielectric losses for a microstrip and a CPW with ws = 100 μm,d = 100 μm, H = 127 μm, and er=11 9. . The dielectrics are characterized with a loss tangent of tan( )d =0 005. . The result is validated with CST.
FIGURE 5.
The output of the proposed tool for the real part ( / )b b0 of the complexnormalized wavenumber for a microstrip with ws = 100 μm, H = 127 μm, and . ,
11 9
r
e = and a CPW with ws = 100 μm, d = 100 μm, H =
3
, and er=11 9. . The result is validated with full-wave CST simulations.β /β0 20 40 60 80 100 2.5 2.6 2.7 2.8 2.9 3 Frequency (GHz)
Phase Constant-Microstrip and CPW Tool CST CPW Microstrip 20 40 60 80 100 0.25 0.3 0.35 0.4 0.45 Frequency (GHz) Loss (dB/ λ0 )
Dielectric Losses-Microstrip and CPW
CPW Microstrip Tool
distribution using the phase constant of the propagating mode:
e ( ) , rj 2 mod 1 2 b b b = +
where b1 and b2 are the phase
con-stants of the adjacent media. The cur-rent ratio rj is then used to define an
equivalent high-frequency resistance
RlRF=r Rj RF [10].
For slot-type transmission lines, because the IE is different, the conduc-tor losses can be calculated by follow-ing the approach proposed in [12]. The approach is based on applying the equiv-alence principle on the slot region by replacing the slot region with the same lossy conductor as that of the ground planes. The lossy conductor can then be implemented in the GF of the strati-fied media by means of an impedance characterized by 2Zgnd, in series with
the stratification. The factor of two used arises here because in the tool, only the equivalent magnetic current is modeled, whereas in [12] the equivalent electric current is also included.
The conductor losses of a microstrip [Figure 7(a)] and a CPW [Figure 7(b)] are validated with ws=100 mn ,
,
d=100 mn H=127 mn , er= 11 9. ,
t=6 mn and v=4.1
$
10 S/m7 . For the microstrip, both the main conductor and ground plane are nonperfect. In (a), both results are shown for the case in which we neglect (rj = 1) and the case inwhich we account for the influence of the
asymmetric current magnitude on the top and bottom of the main conductor. For the CPW, a good match with full-wave simula-tions is provided. Because the different magnitude of magnetic currents is already accounted for in the GFs of the stratified media, no additional compensation factor is needed, as it is in the strip case.
SURFACE-WAVE EXCITATION LOSSES
When the substrate height of a printed transmission line becomes larger than/ ,4
m the stratification will support an
intrinsic surface wave. As the frequen-cy increases, this surface wave can be excited by the main propagating mode along the transmission line (i.e., when
e mod 1 SFW
b b [2]). At this point, the
software tool encloses the surface-wave
poles in the integration path in the transverse domain, and the wavenum-ber becomes complex because the imaginary part is now associated with surface-wave losses.
The losses due to excitation of the TM0 and TE1 surface waves for a
CPW with ws=100 mn , d=100 mn , ,
H=500 mn and er=11 9. are shown in Figure 8. It can be seen that at approximately 65 GHz, the surface-wave condition is verified, and the TM0
surface wave is excited. Additionally, at approximately 118 GHz, the TE1 surface
wave will be excited.
RADIATION LOSSES
A CPW in the presence of an infinite dielectric half-space, simulating a silicon
FIGURE 8.
Losses associated with the excitation of the first two surface-waves (TM0and TE1) for a CPW with ws = 100 μm, d = 100 μm, H = 500 μm, and er=11 9. . The
result is validated with a full-wave CST simulation.
FIGURE 7.
Conductor losses for (a) a microstrip and (b) a CPW with ws = 100 μm, d = 100 μm, H = 127 μm, er=11 9. , t = 6 μm,and v=4 1 10. · 7 S/m. For the microstrip, both the main conductor and the ground plane have a finite conductivity. The results
are validated with CST and Sonnet.
0.2 0.3 0.4 0.5 0.6 0.7 0.75 Conductor Losses–Microstrip Tool with Current Ratio Correction Tool Without Current Ratio Correction CST—Sonnet 10 25 40 55 70 85 100 Frequency (GHz) (a) Loss (dB/ λ0 ) 10 25 40 55 70 85 100 Frequency (GHz) (b) 0.25 0.35 0.45 0.55 0.65 0.75 Conductor Losses–CPW Loss (dB/ λ0 ) Tool CST Sonnet 60 80 100 120 140 101 100 10–1 10–2 10–3 Surface-Wave Losses Frequency (GHz) Loss (dB/mm) CST Tool
lens, is investigated with dimensions , ws= 100 mn d=100 mn ,H=3, and . . 11 9 r
e = The high-density dielectric
lens above the CPW results in signifi-cant radiation of power in the lens, which can be characterized by the software tool. Because the dominant mode is a
leaky-wave pole, the integra-tion path in the transverse domain is required to cross the branch cuts to integrate over the bottom Riemann sheet. The output of the software tool is shown in Figure 9. It can be seen that at 200 GHz, approxi-mately half of the power is radiated into the silicon after a propagation distance of meff.
SURFACE-WAVE LOSS APPROXIMATION
The surface-wave losses in a CPW can be compared with radiation losses of a CPW into an infinite medium; this comparison is shown in Figure 10. As a reference, conductor losses are also calculated,
wherein the conductors have a finite con-ductivity of v=4 1 10. · 7 S/m and a
finite thickness of t=6 mn . Analyzing
Figure 10, it is clear that when the sur-face wave is excited, the losses are soon comparable to those due to direct radia-tion in the infinite dielectric. This is also verified using full-wave CST simula-tions. When a dielectric slab is larger than md/2 in height, the medium can therefore also be modeled as an infinite dielectric. For this reason, the software tool restricts the height of a dielectric slab to an electrical height of md/ ,2 so that what is accounted for is only a first surface wave appearing in the stratification.
CHARACTERISTIC IMPEDANCE
A last important aspect in the charac-terization of transmission line is deter-mining the characteristic impedance of the line. The characteristic impedance (admittance) of slot (strip)-type transmis-sion lines can be calculated using the residue of the pole associated with the main propagating mode along the trans-mission line (7) [13]: e e / det( (( ) )) . Z Y j A kD k N mod mod 0 0 0 Slot Strip= -l (7) For coplanar lines excited in dif-ferential or common excitation mode, (7) is a vector from where the first ele-ment can be taken as the characteristic impedance. The characteristic imped-ance of a microstrip with ws=100 mn ,
,
H=127 mn and er=11 9. is shown in Figure 11. For leaky lines suffering from losses due to radiation or surface-wave excitation, the definition in (7) can still be used, introducing an imaginary part on the characteristic impedance. Interested readers are referred to [14], in which a validation is given because the use of a complex propagation constant is not standard.
SUPERCONDUCTIVITY
In the software tool, the superconduc-tive phenomenon is also implemented by describing a conductor with a kinetic inductance. This makes the tool suitable for characterizing more complex struc-tures such as superconductive feeding networks or resonators in several types
FIGURE 10.
The loss comparison between a CPW printed onto a finite (H = 500 μm) and infinite (H = ∞) dielectric slab with ws = 100 μm, d = 100 μm, and er=11 9. .In addition, ohmic losses are calculated wherein the conductors have a finite conductivity of v=4 1 10. · 7 S/m and thickness t = 6 μm.
FIGURE 9.
The radiation loss for a CPW with ws = 100 μm, d = 100 μm, H = ∞, and. . 11 9
r
e = The result is validated with a full-wave CST simulation.
50 100 150 200 0 1 2 3 Radiation Losses–CPW Frequency (GHz) Tool CST Loss (dB/ λeff ) 20 40 60 80 100 120 140 Frequency (GHz) 10–3 10–2 10–1 100 101 Loss (dB/mm) Loss Comparison–CPW SFW-CST SFW-MATLAB Infinite Slab Ohmic Losses
The software tool also
provides the imaginary
part of the characteristic
impedance when the
investigated lines support
significant radiative losses.
of terahertz detectors, e.g., transition edge sensors [15] or kinetic inductance detectors [16]. The total current flow-ing on the superconductor is the super-position of a current exhibiting normal losses and a lossless superconductive current. This assumption is referred to as the two-fluid model [17]. The soft-ware tool uses a simplified model, wherein the superconductors can be characterized via a complex conductiv-ity v=v1+jv2 [18]. The real part of
the complex conductivity v1 represents
the normal fluid conductivity and the imaginary part v2 represents the super
fluid conductivity. The complex conduc-tivity is introduced in Zs as is studied
in [19]. The surface impedance Zs used
for the impedance boundary condition for a superconductor at low tempera-ture (v1%v2) can be characterized by Zs=Rs+jXs, where the surface resistance can be calculated by
R 1 2 s L 22 1 m v v =
and Xs=~n m0 L=~Ls.mL is the
Lon-don penetration depth, which is related to v2: . 1 L 0 2 m n v ~ =
In the software tool, the surface impedance can be given by providing a
value for the kinetic sheet inductance Ls
and the real part of the conductivity. As an example, we study the effect of superconductivity on the character-istic impedance and propagation con-stant of a microstrip with ws=1 5 m. n ,
,
H=5 mn and er=11 9. . The conduc-tors are considered to be NbTiN with a sheet inductance of Ls = 0.45 pH/sq.
The comparison is shown in Figure 12. It is clear that using a superconduc-tive material can have a significant influence on the characteristic imped-ance of a transmission line as well the propagation constant. No analytical or equivalent formulas can estimate these effects on the propagation constant and characteristic impedance of a trans-mission line; one has to resort to using full-wave simulators.
CONCLUSIONS
To date, no analytical tools exist for the fast characterization of transmission lines in terms of both ohmic and radiative losses. For approximating these phe-nomena, a few equivalent formulas exist. However, they only cover a small subset of transmission lines. A front-end design-er has to resort to full-wave simulations, which are time consuming and require expensive licenses. A software tool is proposed that allows for a quick eval-uation of the losses and impedance matching. The quasi-analytical model used allows for the characterization of several interesting phenomena such as surface-wave excitation, radiation, and superconductivity.
The software tool follows a quasi-analytical approach that is based on transmission line formalism result-ing from a spectral representation of the integral equations. Solving the dis-persion equation produces a complex wavenumber of a specific propagating mode along the transmission line. The complex wavenumber gives informa-tion regarding the dispersion and losses of this mode traveling along the trans-mission line. Subsequently, the quasi-analytical model is tested by validating the transmission line characteristics with full-wave simulations. The tool also pro-vides the imaginary part of the charac-teristic impedance when the investigated
20 40 60 80 100 Frequency (GHz) 0 20 40 60 Z0 (Ω ) Characteristic Impedance Tool Sonnet Re(Z ) Im(Z ) } }
FIGURE 11.
The characteristic impedance for a microstrip with ws = 100 μm,H = 127 μm, and er=11 9. . The characteristic impedance is calculated using (7).
Frequency (GHz) 2.5 2.75 3 3.25 3.5 k/ k0 Propagation Constant Model Sonnet 70 75 80 85 90 Characteristic Impedance 0.5 1.0 1.5 2.0 2.5 Frequency (GHz) (a) (b) 0.5 1.0 1.5 2.0 2.5 PEC Ls = 0.45 pH/sq L s = 0.45 pH/sq PEC Z 0 (Ω ) Model Sonnet
FIGURE 12.
The effect of using a superconductor with Ls = 0.45 pH/sq isinvestigated in the (a) propagation constant and (b) characteristic impedance of a microstrip with ws = 1.5 μm, H = 5 μm, and er=11 9. . The results are validated
lines support significant radiative losses. Interested readers are referred to [14] for a more thorough discussion on this complex characteristic impedance. As mentioned previously, the software tool is freely accessible from our group’s web-site at http://terahertz.tudelft.nl/.
AUTHOR INFORMATION
Sven L. van Berkel (s.l.vanberkel@ tudelft.nl) received his B.S. and M.S. degrees in electrical engineering from the Delft University of Technology (TU Delft), The Netherlands, in 2012 and 2015, respectively. He is working toward his Ph.D. degree in the Tera-hertz Sensing Group, Department of Microelectronics, TU Delft. His research interests include passive imaging systems, ultrawideband anten-nas, and analytical/numerical methods for transmission line characterization. He is a Student Member of the IEEE.
Alessandro Garufo (A.Garufo@ tudelft.nl) received his M.S. degree in telecommunications engineering from the University of Siena, Italy, in 2012. He is working toward his Ph.D. degree in the Tetrahertz (THz) Sensing Group, Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands. His research interests include analytical and numerical meth-ods for antennas and transmission line characterization; modeling and design of THz sources based on photoconduc-tors; and design of antennas, dielectric lens antennas, and antenna arrays. He is a Student Member of the IEEE.
Nuria Llombart (n.llombartjuan@ tudelft.nl) received her M.S. and Ph.D. degrees in electrical engineering from the Polytechnic University of Valencia, Spain, in 2002 and 2006, respectively. She is an associate professor with the Tetrahertz (THz) Sensing Group, Delft University of Technology, The Nether-lands. She has coauthored more than 150 journal and international confer-ence contributions. Her research inter-ests include the analysis and design of planar antennas, periodic structures, reflector antennas, lens antennas, and waveguide structures, with emphasis in the THz range. She was a corecipient of
the H.A. Wheeler Award for the Best Applications Paper of the Year 2008 in
IEEE Transactions on Antennas and Propagation, the 2014 THz Science and
Technology Best Paper Award of the IEEE Microwave Theory and Tech-niques Society, and several NASA awards. She also received the 2014 IEEE Antennas and Propagation Soci-ety Lot Shafai Mid-Career Distin-guished Achievement Award. She is a Senior Member of the IEEE.
Andrea Neto (A.Neto@tudelft.) re -ceived his Laurea degree in electronic engineering from the University of Flor-ence, Italy, in 1994, and his Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2000. He is a full pro-fessor of applied electromagnetism in the Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands, where he formed and leads the Tetrahertz (THz) Sensing Group. His research interests include the analysis and design of antennas, with emphasis on arrays, dielectric lens antennas, wideband antennas, electromagnetic band-gap structures, and THz antennas. He was a corecipient of the H.A. Wheeler Award for the Best Applications Paper of the Year 2008 in IEEE Transactions on
Antennas and Propagation, the Best
Innovative Paper Prize at the 30th Euro-pean Space Agency Antenna Workshop in 2008, and the Best Antenna Theory Paper Prize at the European Conference on Antennas and Propagation in 2010. He is a Fellow of the IEEE.
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